In other words, any binary operation * on S is a rule that assigns to each ordered pair of elements of S a unique element of S .

**Binary
Operations**

**Definitions**

The basic arithmetic operations ℝ on are **addition** (+)**,** **subtraction** (−), **multiplication** (×)**,** and **division **(**÷**). Eminent mathematicians
of the latter part of 19^{th} century and in 20^{th} century
like Abel, Cayley,** **Cauchy, and others,
tried to generalize the properties satisfied by these usual arithmetic
operations. To this end they developed new abstract algebraic structures
through the **axiomatic approach**.
This new branch of algebra dealing with these abstract algebraic structures is
known as **abstract
algebra.**

To begin with, consider a simple example involving the basic
usual arithmetic operations addition and multiplication of any two natural
numbers.

*
m *+* n *∈ ℕ ; *m* ×
*n* ∈ ℕ, ∀*m*, *n* ∈ ℕ = {1, 2, 3,...}

Each of the above two operations yields the following
observations:

(1) At a
time exactly two elements of ℕ are
processed.

(2) The
resulting element (outcome) is also an element of ℕ.

Any such operation defined on a nonempty set is called a **binary
operation** or **a binary** **composition **on the
set** **in abstract algebra.

Any operation _{*} defined on a
non-empty set *S* is called a **binary operation**
on *S* if
the following conditions are satisfied:

(i) The operation _{*} must be defined
for each and every ordered pair (*a* , *b*) ∈ *S* × *S* .

(ii) It assigns a **unique** element *a*∗*b* of *S* to every ordered
pair (*a* , *b*) ∈ *S* × *S* .

In other words, any binary operation _{*} on *S* is a rule that assigns to **each ordered
pair** of elements of *S* a **unique**
element of *S* . Also _{*} can
be regarded as a **function** (**mapping**) with input in the Cartesian product *S* × *S*
and the output in *S* .

∗
: *S* × *S*
→
*S* ; ∗( *a* , *b*) =
*a* ∗ *b*
∈ *S*
, where *a* **b* is an unique element.

A binary operation defined by ∗ : *S* × *S* →
*S* ; ∗( *a* , *b*) =
*a* ∗ *b*
∈ *S*
demands that the output *a* ∗*b *must always lie the given set *S* and not in the complement of it. Then
we say that ‘ ∗ is
closed on *S* ’ or* *‘ *S* is **closed**
with respect to ∗
’. This property is known as the **closure property.**

Any non-empty set on which one or more binary operations are
defined is called an **algebraic structure****.**

Another way of defining a binary operation ∗
on *S* is as follows:

∀* a *,* b *∈ *S *,* a*∗*b *is unique and* a *∗* b *∈ *S *.

**Note**

It
follows that every binary operation satisfies the closure property.

**Note**

The
operation ∗ is just
a symbol which may be + , ×, −, ÷ matrix addition, matrix multiplication,
etc. depending on the set on which it is defined.

For
instance, though + and × are binary on ℕ, − is **not** binary operation on ℕ.

To
verify this, consider (3, 4) ∈
ℕ ×
ℕ.

∗ ( *a* , *b*) = −
(3, 4) =
3 −
4 = −1
∉ ℕ.

Hence − is **not binary operation** on ℕ. So ℕ is to be extended to ℤ in order that − becomes binary operation on ℤ. Thus ℤ is closed with respect to +
, ×,
and −
. Thus (ℤ, + , ×, −) is an algebraic structure.

**Observations**

The
binary operation depends on the set on which it is defined.

(a) The
operation – which is **not binary operation** on ℕ but it is binary on ℤ. The set ℕ is extended to include negative
numbers. We call the included set ℤ.

(b) The
operation ÷ on ℤ is
**not binary
operation** on ℤ.
For instance, for (1, 2) ∈
ℤ ×
ℤ, ÷ (1, 2) = 1/2 ∉ ℤ. Hence ℤ has
to be extended further into ℚ.

(c) It
is a known fact that the division by 0 is **not** defined in basic arithmetic. So ÷ is
binary operation on the set ℚ\{0}.
Thus +
, ×,
−
are binary operation on ℚ and
÷ is binary operation ℚ on
\{0}.

Now the
question is regarding the reasons for extending further ℚ to and then from ℚ to **C**.
Accordingly, a number system is needed where not only all the basic arithmetic
operations +, −, ×, ÷ but also to include the roots of
the equations of the form “ *x*^{2} − 2 = 0 ”
and“ *x*^{2} + 1 = 0 ”.

So, in
addition to the existing systems, the collection of irrational numbers and
imaginary numbers (See Chapter 3) are to be adjoined. Consequently ℝ and then **C** are obtained. The biggest number system **C** properly includes all the other number systems ℕ, ℤ, ℚ,
and ℝ as subsets.

**Table12.1**

** **

Examine
the binary operation (closure property) of the following operations on the
respective sets (if it is not, make it binary):

(i) a ∗* b *= *a* + 3*ab* − 5*b*^{2} ; ∀ *a,b*
∈ ℤ

(i) Since
×
is binary operation on ℤ, *a *,*
b *∈ ℤ ⇒* a *×* b *=* ab *∈* **ℤ** *and* b *×* b *=* b*^{2}* *∈ ℤ ... (1)

The fact
that + is binary operation on ℤ and
(1) ⇒ 3*ab = ( ab + ab + ab) *∈ ℤ and

5*b*^{2} =* *(*b*^{2}* *+*
b*^{2}* *+* b*^{2}* *+* b*^{2}* *+*
b*^{2})* *∈ ℤ. .... (2)

Also a ∈ ℤ and 3ab ∈
ℤ implies *a* + 3*ab* ∈ ℤ. ... (3)

(2), (3),
the closure property of − on ℤ yield
a ∗* b *= ( *a* + 3*ab* − 5*b*^{2} ) ∈ ℤ. Since *a **∗ **b* belongs to ℤ, *
is a binary operation on ℤ.

(ii) In
this problem *a* ∗*b* is in the quotient form. Since the division
by 0 is undefined, the denominator *b* -1must
be nonzero.

It is
clear that *b* −
1 =
0 if *b* = 1. As 1∈ ℚ, ∗
is not a binary operation on the whole of ℚ.

However
it can be found that by omitting 1 from ℚ, the output *a* ∗*b* exists in ℚ\{1} .

Hence ∗ is a binary operation on \{1} .

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